De Morgan's Laws (Set Theory)/Relative Complement
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Theorem
Let $S, T_1, T_2$ be sets such that $T_1, T_2$ are both subsets of $S$.
Then, using the notation of the relative complement:
Complement of Intersection
- $\relcomp S {T_1 \cap T_2} = \relcomp S {T_1} \cup \relcomp S {T_2}$
Complement of Union
- $\relcomp S {T_1 \cup T_2} = \relcomp S {T_1} \cap \relcomp S {T_2}$
General Case
Let $S$ be a set.
Let $T$ be a subset of $S$.
Let $\powerset T$ be the power set of $T$.
Let $\mathbb T \subseteq \powerset T$.
Then:
Complement of Intersection
- $\ds \relcomp S {\bigcap \mathbb T} = \bigcup_{H \mathop \in \mathbb T} \relcomp S H$
Complement of Union
- $\ds \relcomp S {\bigcup \mathbb T} = \bigcap_{H \mathop \in \mathbb T} \relcomp S H$
Family of Sets
Let $S$ be a set.
Let $\family {S_i}_{i \mathop \in I}$ be a family of subsets of $S$.
Then:
Complement of Intersection
- $\ds \relcomp S {\bigcap_{i \mathop \in I} S_i} = \bigcup_{i \mathop \in I} \relcomp S {S_i}$
Complement of Union
- $\ds \relcomp S {\bigcup_{i \mathop \in I} S_i} = \bigcap_{i \mathop \in I} \relcomp S {S_i}$
Source of Name
This entry was named for Augustus De Morgan.